= Guinier expansion of a polymer scattering function
{c}
{title2=$g(k)/N=1-k^2R_g^2/3+O(k^4)$}
Expand the isotropic angular factor as $\sin(kr)/(kr)=1-k^2r^2/6+O(k^4r^4)$. For a <polymer scattering function> and finite fourth pair moments, the constant double sum gives $g(0)=N$, and the quadratic sum is determined by the <radius of gyration>:
$$
\frac{g(k)}N=1-\frac{k^2R_g^2}{3}+O(k^4).
$$
In the range $kR_g\ll1$, this is also approximated by $e^{-k^2R_g^2/3}$. The coefficient follows from isotropy and the definition of $R_g$ and does not require a <Gaussian chain>.
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